Problem: Solve for $x$ and $y$ by deriving an expression for $x$ from the second equation, and substituting it back into the first equation. $\begin{align*}-4x-3y &= 1 \\ -5x-2y &= -3\end{align*}$
Solution: Begin by moving the $y$ -term in the second equation to the right side of the equation. $-5x = 2y-3$ Divide both sides by $-5$ to isolate $x$ $x = {-\dfrac{2}{5}y + \dfrac{3}{5}}$ Substitute this expression for $x$ in the first equation. $-4({-\dfrac{2}{5}y + \dfrac{3}{5}}) - 3y = 1$ $\dfrac{8}{5}y - \dfrac{12}{5} - 3y = 1$ Simplify by combining terms, then solve for $y$ $-\dfrac{7}{5}y - \dfrac{12}{5} = 1$ $-\dfrac{7}{5}y = \dfrac{17}{5}$ $y = -\dfrac{17}{7}$ Substitute $-\dfrac{17}{7}$ for $y$ in the top equation. $-4x-3( -\dfrac{17}{7}) = 1$ $-4x+\dfrac{51}{7} = 1$ $-4x = -\dfrac{44}{7}$ $x = \dfrac{11}{7}$ The solution is $\enspace x = \dfrac{11}{7}, \enspace y = -\dfrac{17}{7}$.